In our first department meeting of the year, we worked on a puzzle.
There are 243 coins on a table. 107 of them are face up.
You are blindfolded. You need to divide up the 243 coins into two piles (they don’t have to be equal sizes) and then somehow get the number of face up coins in each pile to be the same. “IMPOSSIBLE!” you say, “there are an odd number of coins face up!” I say, “Wait! There’s more! You’re allowed to flip coins over. You’re blindfolded, so you don’t know which coins your flipping over, but you’re allowed to.”
We were told it was possible.
No one got the answer after 15 minutes, but we made some progress. Then we moved on with the meeting.
I went to lunch with my department head, and we thought we solved the problem. And we determined that our solution would work if the number of coins on the table was divisible by three (it turns out in hindsight that I think we were wrong about our method). We texted the teacher who posed the problem to ask if the number of coins had to be a multiple of three, and he said it was possible no matter how many coins are on the table.
So over sushi, using coins from my department head’s purse, and no paper and no pencil, we came up with a general solution. We were pretty sure it would work, but I did the actual mathematical part of it to check, and lo and behold it does.
This, obviously, is one good thing.